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Theorem domnnzr 19116
 Description: A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Assertion
Ref Expression
domnnzr (𝑅 ∈ Domn → 𝑅 ∈ NzRing)

Proof of Theorem domnnzr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2610 . . 3 (.r𝑅) = (.r𝑅)
3 eqid 2610 . . 3 (0g𝑅) = (0g𝑅)
41, 2, 3isdomn 19115 . 2 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = (0g𝑅) → (𝑥 = (0g𝑅) ∨ 𝑦 = (0g𝑅)))))
54simplbi 475 1 (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  0gc0g 15923  NzRingcnzr 19078  Domncdomn 19101 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-domn 19105 This theorem is referenced by:  domnring  19117  opprdomn  19122  abvn0b  19123  fidomndrng  19128  domnchr  19699  znidomb  19729  nrgdomn  22285  ply1domn  23687  fta1glem1  23729  fta1glem2  23730  fta1b  23733  lgsqrlem4  24874  idomrootle  36792  deg1mhm  36804
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